Weak Among the Strong: Imperfect Information and Perfect Play
Part 2 - Retrograde Analysis and Virtual InformationChad Ellis

Last week we began to look at the strategic implications of playing a game with imperfect information, including the opportunities imperfect information provides for bluffing. This week we'll look at moments where one or both players have perfect or virtually perfect information and can therefore put judgment calls aside and strive for perfect play.

Let's start with a simple example. You are at four life and your board consists of four 1/1 squirrel tokens and a Krosan Avenger. You have several lands in play and six cards in your graveyard (none with flashback) and two lands in your hand. Your opponent has three life, no cards in hand, lots of untapped land (including access to W, U and G) and his board consists of three bear tokens. He has four cards in his graveyard, none with flashback. On his last turn he played a land and said go. (On your turn before that, you played a Krosan Avenger and also said go.)

What do you do? You simply attack with all of your creatures and say a quiet thank-you to whomever it was at WotC that decided that Krosan Avenger should have trample. No matter how your opponent blocks he takes at least three points of damage and he has no other possibilities. Because of perfect information (IE no cards in his hand), you can make a perfect play.

But what if he has three cards in hand? Now the simple answer gets tricky.

You could alpha-strike and will win if your opponent has nothing, but you will lose if he has virtually any trick. Maindeckable commons that will cost you the game if you go for the win include Embolden, Muscle Burst, Second Thoughts, Chastise, Moment's Peace, Guided Strike, Prismatic Strands, Aether Burst, Repel, Pay No Heed and Gallantry. That's quite a list.

What happens if you sit back? If your opponent has nothing, attacking is pretty risky for him, since you can wreck him with any number of tricks as well. Even playing a random spell would give the Avenger the ability to regenerate, so you could trade four squirrels for three bears and be left with the only creature in play. So perhaps you should sit back for a turn. After all, if you draw so much as an Obsessive Search you can start sending your Avenger and killing his Bears off, while keeping back four blockers to avoid losing to a trick. He didn't attack last turn; maybe he'll keep waiting.

This isn't horrible reasoning, and my experience is that this is where most players stop. Some will attack, some will pass the turn, but they will all have the same rough starting point of, "Well, an alpha strike might win, but I lose if he has any of these cards." Now let's look at how to do better.

First, let's look at the above list. Is his last turn inconsistent with any of those cards? Suppose he was holding Embolden. Unless he's a very cautious player, he would probably have attacked you last turn with all three of his bears, since Embolden will probably save his attackers and keep him alive when you counter attack next turn. Play through the natural blocks you might make, from chumping one bear with a squirrel or blocking one bear with a regenerating Avenger to putting four squirrels in front of two bears with or without the Avenger blocking as well, and you'll see what a good job Embolden does of dominating this particular combat. So at the very least you can probably conclude that he's not holding Embolden.

This is an extremely important form of retrograde analysis! Many players are good at positive analysis, by which I mean making positive assessments about cards their opponent has (or is at least representing). Most spend less effort, and are therefore less skilled, at the trickier business of negative analysis.

Sometimes retrograde analysis, both positive and especially negative, can look back over several turns. Does your opponent have any kind of counterspell? Two turns ago you cast Flametongue Kavu and killed his Lightning Angel. He did nothing during his next turn and you attacked, putting him at three. He drew for his turn and cast Wrath of God, leaving him with two lands untapped, and passed the turn.

So what do you know? It is very likely that he didn't have a counterspell when you played Flametongue Kavu. It's also very unlikely that he held on to Wrath of God for a turn while you knocked him from seven to three. So that means he had no counters before drawing two cards, and one of those cards was Wrath.

Now instead of looking at his hand of three cards and wondering what the odds are that one of them is a counterspell, you only have to determine what the odds are of one draw being a counterspell. And you aren't done yet! Suppose you know his decklist is a card-for-card copy of Brian Kibler's Miss America deck, except with Wrath of God instead of Opposition. (N.B. I do NOT advocate this swap at all…I'm merely looking for an easy example with a decklist readers can link to.)

Now let's examine the mystery draw. It wasn't a creature; he would have cast it. It wasn't Wrath of God, Repulse, Fire/Ice, or Goblin Trenches. Any of them would have been cast rather than taking a hit from the Flametongue when at seven life.

At this point we have eliminated every spell other than Memory Lapse!

Can we do better?

Remember, we said that he did nothing during the turn of the mystery draw, and that he has three cards in hand. Since he has drawn once and cast Wrath of God since then, he had three cards in hand at that point. None of them are spells either, since we've already concluded that he would have Lapsed the Flametongue if possible and he would have cast any other spell immediately, as we've already determined. That means that his original two cards were both land and if he drew a land he chose to keep three lands in hand for bluff value rather than play one of them!

How likely is this? Not very. Miss America has all sorts of draws where it wants its land in play and its spell base doesn't let it gain much by holding lots of land to bluff. If he doesn't know that you know his decklist, he might hold a land or two back in order to make you think he had something. But no good player, and especially no aspiring Kibler, is likely to hold three lands in hand with just six in play while staring at a Flametongue Kavu and knowing that his next draw might be Deep Analysis. He'd play a seventh land, knowing that if his next draw is Deep Analysis and he draws Wrath off of that, he's still in the game. (In fact, I think this analysis is enough to suggest that your opponent is trying too hard to bluff you; he should at most have held one land for bluffing and quite possibly not even that!)

So, the mystery draw wasn't a spell other than Memory Lapse, and it wasn't land. That means it was Memory Lapse. The other two cards in his hand are land, at least one of which really should be in play.

Welcome to perfect information.

Actually, at this point I really should say, "Welcome to the point where you go over your analysis." Sometimes what you will do based on imperfect information is radically different from what you will do if your information is perfect. When the information is critical, it's especially valuable to take the time to reconsider your assumptions.

Virtual information

One of my favorite theoretical concepts in discussing card advantage is virtual card advantage. (I believe the concept originated with EDT.) Virtual card advantage exists when one or more of your cards neutralize a larger number of your opponent's cards. An Engineered Plague on squirrels doesn't just kill squirrels in play. It also "kills" any Nests or Chatters your opponent is holding or draws, as well as weakening his Mobs. Similarly, a Hallowed Healer and Mystic Zealot might hold off a medium-sized army. If two creatures are keeping four or five at bay, that is card advantage.

Virtual information works differently. Virtual information is when you don't know what is in your opponent's hand (or in his deck or near the top of yours) but can and should act as though you do.

I doubt there is a reader out there who hasn't done something like the following: You're playing limited and get off to a great start, playing threats on turns two through six. Your opponent struggles a bit, not playing a creature on turn two, or maybe her creatures simply aren't as good as yours. Your advantage builds turn by turn, and after a Muscle Burst during your attack on turn seven took out two gang-blockers while leaving your attacker alive, the game is looking like it could be a rout. Triumphantly you play the land you drew for the turn and cast the Rampant Elephant in your hand, knowing that nothing can save her now.

Then you look at the board and at the two cards in her hand and start to panic. What if she has Rout or Mutilate? She's got two cards and is about to draw a third, while you're completely out of gas...

There may be no reason to infer Mutilate or Wrath from her play. You might even be able to calculate that she doesn't have it. Heck, maybe she doesn't have any cards in hand. Putting out your last threat is most likely still a mistake. Why? Because, barring something gross, you are winning whether you play the Elephant or not.

Think of this as a two-by-two matrix. (Forgive me, I'm an MBA, remember?) The vertical axis is "Play Elephant" and "Don't Play Elephant." The horizontal axis is "She has mass removal" and "She doesn't have mass removal."

She has mass removal

She doesn't

Play Elephant

Wrong Play

Win

Don't Play Elephant

Right Play

Win

Clearly if she has mass removal you would be better off not playing the Elephant. But look at the right column. If she doesn't have mass removal it doesn't matter at all what you do. Your advantage on the board is so great that even if she plays out a creature or two, you are going to finish her.

Under these conditions you have access to the virtual information that she is going to cast a mass removal spell. Since your decision doesn't matter otherwise, you simply assume that she does and play accordingly.

Sometimes virtual information will mean concluding that your opponent doesn't have a particular spell or class of spells. It may be that in the matrix above you will lose regardless to a Wrath but that playing out your hand is clearly better if she doesn't have it. Again, you simply eliminate the column where your decision doesn't matter and act as though you know she doesn't have it.

Virtual information can also apply to the card(s) you are about to draw. A disappointing example comes from Grand Prix Manchester, where my RBu deck was facing a Cloaked Llanowar Knight. I was one turn from dying when I drew Zap. Clearly I'm going to cast it, but at what target? My only relevant topdeck option was Urza's Rage, since my three bounce spells were all Recoils. The "Top Card Isn't Rage" column got deleted and I clearly had to Zap the Knight. (Alternately, I could delete the "Zap Opponent" row, since I lose either way if I do that.) Alas, I didn't even get to draw a card, as my Zap was Confounded. My opponent had access to the same virtual knowledge that I did and correctly played as though the Rage was my next card.

Now let's complicate the issue slightly. We'll add a Viashino Grappler to his side of the table and we'll put me at four life. We'll also decide that I've got a Repulse in my deck, as well as a Scorching Lava. Now the matrix looks like this:

Next Card is:

Urza's Rage

Repulse

Scorching Lava

Other

Zap Grappler

Wrong

Right

Die Next Turn

Die Next Turn

Zap Knight

Right

Wrong

Die Next Turn

Die Next Turn

Zap Opponent

Wrong

Wrong

Die Next Turn

Die Next Turn

Here we can eliminate the second two columns; so we now have the virtual information that our next card is either Rage or Repulse. We then eliminate the bottom row, since it is wrong regardless. Now we have a two-by-two matrix with no clear answer:

Urza's Rage

Repulse

Zap Grappler

Wrong

Right

Zap Knight

Right

Wrong

We still work on making the perfect decision. If I had two copies of Repulse, that would speak strongly in favor of Zapping the Grappler, since I would have twice as good a chance of surviving the next turn. But let's assume I've got just one copy of each. Since each outcome is equally likely, we can ask which situation is better if I guess correctly. That is, am I better off Zapping the Grappler and drawing Repulse or Zapping the Knight and drawing Rage?

All things being equal, the first option looks much better to me. Instead of a Grappler on the board and me going quickly to one life, my opponent will have a Knight in his hand. I'll also have drawn another card into my deck.

What happens if I have a Slimy Kavu in my hand? First of all, we have to reinstate the Scorching Lava and potentially the "other" column, since now I have a chump-blocker. Now if I Zap the Grappler and draw Scorching Lava I can cast the Kavu to block. Sadly I don't have mana to finish the Knight off, but if my next card is Rage I at least have a chance.

The Slimy Kavu also makes the Zap/Repulse play much better than Zap/Rage. Killing the Knight and going to one in order to play a blocker the Grappler can trample over for the kill isn't much fun. Killing the Grappler and Repulsing the Knight is much better. The Slimy can then trade with the Knight after it comes out. Four life is better than nothing.

He didn't call it by name, but Paul Sottosanti wrote a terrific article for StarCity that describes some practical examples of virtual information. In one case, Jeff Fung made a very reasonable play, saving a Wild Mongrel with a Muscle Burst. Fung was going to die unless he drew a Fiery Temper on his next turn. As Sottosanti explains, however, if Fung had let the Mongrel go there would actually have been two topdecks that would save him (Pardic Firecat and the Sonic Seizure he actually drew). Fung used virtual knowledge, but his analysis was incomplete. His full virtual knowledge should have been that his next card was one of those three, in which case he would no doubt have decided to make the decision that was twice as likely to win for him. In another example, Sottosanti explains how he put himself in a position to lose games two and three of a match if his opponent cast Upheaval, a card Sottsanti knew he had. He got lucky in game two but lost game three because he failed to act on the available virtual information.